Integral averaging technique for the interval oscillation criteria of certain second-order nonlinear differential equations

Abstract

We present new interval oscillation criteria related to integral averaging technique for certain classes of second-order nonlinear differential equations which are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [t 0 , ∞), rather than on the whole half-line. They generalize and improve some known results. Examples are also given to illustrate the importance of our results.  2004 Elsevier Inc. All rights reserved.

Oscillation of second order differential equations with mixed nonlinearities

“…(1.1)-(1.2). Our results generalize and improve some known results in [4,13,15,16,20,22]. Finally, several examples are also given to illustrate the importance of our results.…”

“…Therefore, the problem is to find oscillation criteria that use only the information about the involved functions on these intervals; outside of these intervals the behavior of the functions is irrelevant. Such type of oscillation criteria are referred to as interval oscillation criteria, see, e.g., [4,20,22].…”

The Kamenev type interval oscillation criteria of mixed nonlinear impulsive differential equations under variable delay effects

“…Assume that the conditions of Lemma 2.1 hold. Let x(t) be a positive solution of (1) and u(t) be defined by (7). Then, for any (H 1 , H 2 ) ∈ H , we have…”

“…, σ (t) ∈ C([t 0 , ∞)) and there exists a nonnegative constant σ 0 such that 0 ≤ σ (t) ≤ σ 0 for all t ≥ t 0 , r(t) ∈ C 1 ([t 0 , ∞), (0, ∞)) is nondecreasing. For some particular cases of (1), many authors have devoted work to the interval oscillation problem (see [3][4][5][6][7][8][9][10][11][12][13]). Particularly, when α = 1, a k = b k = 1 and σ (t) = 0, (1) reduces to the mixed type Emden-Fowler equation…”

Interval oscillation criteria for second-order forced impulsive differential equations with mixed nonlinearities

“…In 1999, Wong [9] substantially improved the results of El-Sayed with a more direct and simpler proof. Later, further development of the "interval criteria" for oscillation have been obtained by many authors for both differential equations and delay differential equations in several directions, see [10][11][12][13][14][15][16].…”